reserve V for RealLinearSpace;

theorem Th17:
  for v being VECTOR of V, X being Subspace of V, y being VECTOR
of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X for
w being VECTOR of X + Lin{v}, x being VECTOR of X,
  t,r being Real st w |-- (W,
  Lin{y}) = [x,r*v] holds t*w |-- (W,Lin{y}) = [t*x, t*r*v]
proof
  let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be
  Subspace of X + Lin{v};
  assume that
A1: v = y and
A2: X = W and
A3: not v in X;
A4: X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,A2,A3,Th11;
  let w be VECTOR of X + Lin{v}, x be VECTOR of X,
      t,r be Real such that
A5: w |-- (W,Lin{y}) = [x,r*v];
  reconsider z = x as VECTOR of X + Lin{v} by A2,RLSUB_1:10;
  r*y = r*v by A1,RLSUB_1:14;
  then
A6: t*w = t*(z + r*y) by A4,A5,Th3
    .= t*z + t*(r*y) by RLVECT_1:def 5
    .= t*z + t*r*y by RLVECT_1:def 7;
  reconsider u = x as VECTOR of V by RLSUB_1:10;
A7: t*r*y in Lin{y} by RLVECT_4:8;
A8: t*r*y = t*r*v by A1,RLSUB_1:14;
A9: t*z = t*u by RLSUB_1:14
    .= t*x by RLSUB_1:14;
  then t*z in W by A2;
  hence thesis by A4,A9,A8,A7,A6,Th2;
end;
